 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2017/s1/1/2/solution

Choose a continuous argument

θ

on a simply connected domain avoiding zero, and put

r = x^{2} + y^{2}

. With

u = x lo g r - y θ

u_{x} = 1 + lo g r, u_{y} = - θ, u_{xx} = x / r^{2}, u_{yy} = - x / r^{2} .

Here

θ_{x} = - y / r^{2}

θ_{y} = x / r^{2}

; these identities specify the branch needed in the written arctangent. Integrating the Cauchy-Riemann equations gives

f (z) = z Log z + i C, v = y lo g r + x θ + C, C \in R .

The literal principal

arctan (y / x)

works only separately on suitable half-planes. It cannot stand for a single continuous argument across the imaginary axis.

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